Moment of Force Dimensional Formula: Fundamental Dimensions Explained

Correct Answer: (D) ML²T⁻²

Understanding the Moment of Force

The moment of force, also known as torque, describes the turning effect produced by a force about a fixed point, axis, or pivot. When a force is applied to an object at some perpendicular distance from its axis of rotation, the force tends to rotate the object. This rotational effect is called the moment of force.

A familiar example is opening a door. When force is applied near the outer edge of the door, far from the hinges, the door opens more easily. However, applying the same force close to the hinges produces a much smaller turning effect. This shows that the moment of force depends not only on the magnitude of the applied force but also on the perpendicular distance between the axis of rotation and the line of action of the force.

Formula for the Moment of Force

The magnitude of the moment of force is mathematically expressed as:

Moment of force = Force × Perpendicular distance

Therefore,

τ = F × r

where τ represents the moment of force or torque, F represents the applied force, and r represents the perpendicular distance from the axis of rotation to the line of action of the force.

To determine the fundamental dimensions of the moment of force, we need to find the dimensions of force and distance separately and then multiply them.

Step-by-Step Derivation of the Dimensional Formula

Step 1: Find the Dimensions of Force

According to Newton’s second law of motion, force is equal to the product of mass and acceleration:

Force = Mass × Acceleration

The fundamental dimension of mass is M. Acceleration is the rate of change of velocity with time. Since velocity has dimensions LT⁻¹, acceleration has dimensions LT⁻².

Therefore, the dimensional formula of force is:

[F] = M × LT⁻²

Hence,

[F] = MLT⁻²

Step 2: Find the Dimensions of Distance

The perpendicular distance involved in the formula for the moment of force is a length quantity. Therefore, its fundamental dimension is:

[r] = L

Step 3: Substitute the Dimensions into the Formula

We know that:

Moment of force = Force × Distance

Substituting the dimensional formulas of force and distance gives:

[τ] = [MLT⁻²] × [L]

Combining the powers of length:

[τ] = ML²T⁻²

Therefore, the fundamental dimensions of the moment of force are ML²T⁻². Hence, option (D) is the correct answer.

Detailed Analysis of Each Option

Option (A): MLT⁻¹

Option (A) is incorrect because MLT⁻¹ represents the dimensional formula of linear momentum. Linear momentum is defined as the product of mass and velocity. Since mass has dimension M and velocity has dimension LT⁻¹, the dimensional formula of momentum becomes MLT⁻¹. The moment of force, however, contains an additional length factor and has a time exponent of −2 rather than −1.

Option (B): MLT⁻²

Option (B) is also incorrect because MLT⁻² is the dimensional formula of force. The moment of force is not simply a force; it is the product of force and perpendicular distance. Therefore, the dimensional formula of force must be multiplied by one additional length dimension L. This changes MLT⁻² into ML²T⁻².

Option (C): ML⁻¹T⁻¹

Option (C) is incorrect because the negative power of length does not arise in the formula for the moment of force. Torque is obtained by multiplying force by distance, so the power of length must increase rather than decrease. Since force already contains one power of length and is multiplied by another length, the final power of L must be 2.

Option (D): ML²T⁻²

Option (D) is correct. The moment of force is the product of force and perpendicular distance. The dimensional formula of force is MLT⁻², while the dimensional formula of distance is L. Multiplying them gives ML²T⁻², which is the required dimensional formula of the moment of force.

Relationship Between Moment of Force and Torque

The terms moment of force and torque are often used interchangeably in elementary physics. Both describe the rotational effect produced by a force. In vector form, torque is written as the cross product of the position vector and force:

τ = r × F

Its magnitude can be written as:

τ = rF sin θ

where θ is the angle between the position vector and the applied force. Since sin θ is a dimensionless quantity, it does not affect the dimensional formula. Therefore, the dimensions of torque remain equal to the dimensions of force multiplied by length:

[τ] = ML²T⁻²

SI Unit of Moment of Force

The SI unit of force is the newton (N), while the SI unit of perpendicular distance is the metre (m). Therefore, the SI unit of moment of force is:

newton metre (N m)

One newton can be expressed in fundamental SI units as:

1 N = 1 kg m s⁻²

Therefore:

1 N m = 1 kg m² s⁻²

This unit expression directly confirms the dimensional formula:

ML²T⁻²

Why Torque and Energy Have the Same Dimensions

An interesting point in dimensional analysis is that both torque and energy have the same fundamental dimensions, namely ML²T⁻². Work or energy is calculated as force multiplied by displacement, while torque is calculated as force multiplied by perpendicular distance. Because both expressions involve the product of force and length, their dimensional formulas are identical.

However, torque and energy are physically different quantities. Energy is a scalar quantity, whereas torque is a vector quantity associated with rotational motion. Their SI units are also conventionally written differently. Energy is expressed in joules (J), while torque is expressed in newton metres (N m). Therefore, having the same dimensional formula does not necessarily mean that two physical quantities represent the same physical concept.

Final Answer

The moment of force is calculated by multiplying force by the perpendicular distance from the axis of rotation. Since the dimensions of force are MLT⁻² and the dimensions of distance are L, the dimensional formula of the moment of force is:

[Moment of force] = [Force] × [Distance]

= MLT⁻² × L

= ML²T⁻²

Therefore, the correct answer is (D) ML²T⁻².